mathematics iii ts 4353 class b
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Mathematics III TS 4353 Class B. Herlina Setiyaningsih Civil Engineering Department Petra Christian Universit y. Jurusan Teknik Sipil Matematika III (TS 4353) Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1. PD Serentak (Simultan). f 1 (D)y + g 1 (D)z = h 1 (x) - PowerPoint PPT PresentationTRANSCRIPT
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Herlina SetiyaningsihCivil Engineering Department
Petra Christian University
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PD Serentak (Simultan)f1(D)y + g1(D)z = h1(x)
f2(D)y + g2(D)z = h2(x)
PD: ∆y = ∆1 dan ∆z = ∆2
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 3: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/3.jpg)
Example 1PD: Dy – z = ex
y + (D+2)z = 0
Δy = Δ1 (D2 + 2D +1) y= 3ex
yc = e-x(c1+c2x)
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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Example 1 (Lanjutan)
PUPD: y = yc + yp y = e-x (c1 + c2x) + ¾ ex
Δz = Δ2 (D2 + 2D +1)z = -ex
zc = e-x(c3+c4x)
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
PUPD: z = zc + zp z = e-x (c3 + c4x) -1/4ex
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Example 1 (Lanjutan)y dan z masuk ke PD
D[e-x (c1 + c2x) + ¾ ex]- [e-x (c3 + c4x) -1/4ex]=ex
-e-x (c1 + c2x) + e-x c2 + ¾ ex -e-x c3 - e-x c4x + 1/4ex
= ex
e-x x (-c2 - c4) + e-x (-c1 + c2 - c3) = 0
-c2 - c4 = 0 c2 = - c4
-c1 + c2 - c3 = 0 c3 = -c1 + c2
PUPD: y = yc + yp y = e-x (c1 + c2x) + ¾ ex
PUPD: z = zc + zp z = e-x ((-c1 + c2 ) – c2x) -1/4ex
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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Example 2PD: (D+2)y = z
(D+1)z = 2y(D+2)y – z = 0-2y + (D+1)z = 0
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
= D2 + 3D
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Example 2 (Lanjutan)Δy = Δ1 (D2 + 3D) y= 0
yc = c1+c2e-3x
Δz = Δ2 (D2 + 3D) z= 0
zc = c3+c4e-3x
y dan z masuk ke PD(D+2)y – z = 0(D+2) (c1+c2e-3x) – (c3+c4e-3x) = 0
D(c1+c2e-3x) + 2c1 + 2c2e-3x– c3- c4e-3x = 0
0 - 3c2e-3x + 2c1 + 2c2e-3x– c3- c4e-3x = 0
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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e-3x(-3c2+ 2c2 - c4) + (2c1 – c3) = 0
-c2 - c4= 0 c2 = -c4
2c1 – c3 = 0 2c1 = c3
PUPD: y = c1 + c2e-3x
PUPD: z = 2c1- c2e-3x
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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PD Euler(ax+b)2 y” + p(ax+b)y’ + qy = f(x); a, b, p, q
konstanSubstitusi: ax+b = et t = ln (ax+b)
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 11: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/11.jpg)
Example 1(2x+1)2y” – 2(2x+1)y’ – 12y = 6xSubst: 2x+1 = et t=ln(2x+1)
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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Example 1 (Lanjutan)
(4D2 – 8D – 12)y = 3et - 3PR: (4D2 – 8D – 12)y = 0 PK : 4k2 – 8k – 12 = 0
k2 – 2k – 3 = 0 (k-3)(k+1) = 0k = 3 dan k = -1
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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yc = c1 e-t + c2 e3t
PUPD: y = yc + yp
= c1 e-t + c2 e3t -3/16et + ¼
= c1 (2x+1)-1 + c2 (2x+1)3 - 3/16(2x+1) + ¼
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 14: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/14.jpg)
Aplikasi Bidang T. SipilPada struktur dinamik dikenal adanya getaran.
Persamaan umum getaran pada struktur dinamik :
- perpindahan (x)- kecepatan- percepatan
Sistem getaran dibagi menjadi 4:1. Getaran bebas2. Getaran bebas dan teredam3. Getaran terpaksa 4. Getaran teredam & terpaksa
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 15: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/15.jpg)
Sistem getaran dibagi menjadi 4:1. Getaran bebas
2. Getaran bebas dan teredam
3. Getaran terpaksa
4. Getaran teredam & terpaksa
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 16: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/16.jpg)
Getaran Terpaksamy” + ky = Fo cos ωtyc = e0t(c1 cos ω0t + c2 sin ω0t )
yp = B1 cos ωt + B2 sin ωt
y’p = -B1 ω sin ωt + B2 ω cos ωt
y”p = -B1 ω2 cos ωt - B2 ω2 sin ωt
m (-B1 ω2 cos ωt - B2 ω2 sin ωt) + k (B1 cos ωt + B2 sin ωt) = Fo cos ωt
B1 (k-m ω2) cos ωt + B2(k-m ω2) sin ωt = Fo cos ωtB1 (k-m ω2) = Fo B1 = Fo/ k-m ω2 B2(k-m ω2) = 0 Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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PUPD:y = c1 cos ω0t + c2 sin ω0t + (Fo/ k-m ω2)cos ωt
Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
![Page 18: Mathematics III TS 4353 Class B](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56814481550346895db117b9/html5/thumbnails/18.jpg)
Getaran Terpaksa & Teredammy” + cy’ + ky = Fo sin ωtyc = ??
yp = B1 cos ωt + B2 sin ωt
y’p = -B1 ω sin ωt + B2 ω cos ωt
y”p = -B1 ω2 cos ωt - B2 ω2 sin ωt
m(-B1 ω2 cos ωt - B2 ω2 sin ωt)+c(-B1 ω sin ωt + B2 ω cos ωt ) + k (B1 cos ωt + B2 sin ωt) = Fo sin ωt
B1 = [(k-m ω2 ) Fo] / [(k-m ω2)2 + (c ω) 2]B2 = [-c ω Fo] / [(k-m ω2)2 + (c ω) 2]Jurusan Teknik Sipil Matematika III (TS 4353)
Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1
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Finish!!!!Bertemu di Integral
Rangkap